Abstract
A counterpart of the Ohlin theorem for convex set-valued maps is proved. An application of this result to obtain some inclusions related to convex set-valued maps in an alternative unified way is presented. In particular counterparts of the Jensen integral and discrete inequalities, the converse Jensen inequality and the Hermite–Hadamard inequalities are obtained.
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1 Introduction
In Ohlin [12] proved the following interesting and very useful result on convex functions in a probabilistic context (as usual, \({\mathbb {E}}[X]\) denotes the expectation of the random variable X):
Lemma 1
[12]. Let \(X_1\), \(X_2\) be two real valued random variables such that \(\mathbb {E}[X_1]=\mathbb {E}[X_2]\). If the distribution functions \(F_{X_1} , F_{X_2}\) cross one time, i.e. there exists \(t_0 \in {\mathbb R}\) such that
then
for every convex function \(f:{\mathbb R}\rightarrow {\mathbb R}.\)
For years the above Ohlin lemma was not well-known in the mathematical community. It has been rediscovered by Rajba [14], who found its various applications to the theory of functional inequalities. In [13, 15, 18], the Ohlin lemma is used, among others, to get a simple proof of the known Hermite–Hadamard inequalities, as well as to obtaining new Hermite–Hadamard type inequalities.
In this note we prove counterparts of the Ohlin theorem for convex set-valued maps. We present also applications of these results to obtain some inclusions connected with convex set-valued maps.
2 Preliminaries
Let \((Y, \Vert \cdot \Vert )\) be a separable Banach space, B be the closed unit ball in Y, \((\Omega ,\mathcal {A}, P)\) be a probability space with a non-atomic measure P and \(I\subset {\mathbb R}\) be an open interval. Denote by n(Y) the family of all nonempty subsets of Y and by cl(Y) the family of all closed nonempty subsets of Y. For a given set-valued map \(G:\Omega \rightarrow n(Y)\) the integral \(\int _{\Omega } G(\omega ) dP\) is understood in the sense of Aumann, i.e. it is the set of integrals of all integrable (in the sense of Bochner) selections of the map G (cf. [1, 2]):
A set-valued map \(G:\Omega \rightarrow n(Y)\) is called integrable bounded if there exists a non-negative integrable function \(k:\Omega \rightarrow {\mathbb R}\) such that \(G(\omega ) \subset k(\omega ) B\), for all \(\omega \in \Omega .\) In this case every measurable selection of G is integrable and, consequently, the Aumann integral of G is nonempty whenever G is measurable.
The following properties of the Aumann integral will be needed in our investigations:
Lemma 2
[1], Theorems 8.6.3, 8.6.4 ]. Let \(G:\Omega \rightarrow cl(Y)\) be a measurable set-valued map. a) The closure of the integral of G is convex and
b) If Y is finite dimensional, then the integral of G is convex. In particular, if \(Y={\mathbb R}\) and \(G(\omega )=[g_1(\omega ), g_2(\omega )], \omega \in \Omega \), then
c) If G is integrable bounded, then
Recall that a set-valued map \(G:I\rightarrow n(Y)\) is called convex if
for all \(x_1,x_2\in I\) and \(t\in [0,1]\) (see e.g. [1, 3, 4, 8] and the references therein). Note that by (2), all values of G are convex subsets of Y.
The following lemma characterizes convex set-valued maps with values in \(cl({\mathbb R})\).
Lemma 3
[8] . A set-valued map \(G:I\rightarrow cl({\mathbb R})\) is convex if and only if it has one of the following forms:
-
a)
\(G(x)= [g_1(x),g_2(x)], \quad x\in I,\)
-
b)
\(G(x)= [g_1(x),+ \infty ), \quad x\in I,\)
-
c)
\(G(x)= (- \infty ,g_2(x)], \quad x\in I,\)
-
d)
\(G(x)= (-\infty , + \infty ), \quad x\in I,\)
where \(g_1:I\rightarrow {\mathbb R}\) is convex and \(g_2:I\rightarrow {\mathbb R}\) is concave.
Clearly, if \(G:I\rightarrow cl({\mathbb R})\) is convex and integrable bounded, then it is of the form a).
3 Ohlin-Type Result for Convex Set-Valued Maps
The following result is a counterpart the Ohlin lemma for convex set-valued maps.
Theorem 4
Let \((Y, \Vert \cdot \Vert )\) be a separable Banach space, \((\Omega ,\mathcal {A}, P)\) be a probability space with a non-atomic measure P and \(I\subset {\mathbb R}\) be an open interval. Assume that \(X_1, X_2:\Omega \rightarrow I\) are integrable random variables such that \(\mathbb {E}[X_1]=\mathbb {E}[X_2]\). If there exists \(t_0 \in {\mathbb R}\) such that
then
for every convex integrable bounded set-valued map \(G:I\rightarrow cl(Y)\).
Proof
The proof is divided into two steps. First, we assume that \(Y={\mathbb R}\). Then, by Lemma 3 and the assumption that G is integrable bounded, we obtain that G is of the form \(G(x)= [g_1(x),g_2(x)]\), \(x\in I\), where \(g_1:I\rightarrow {\mathbb R}\) is convex and \(g_2:I\rightarrow {\mathbb R}\) is concave. By the Ohlin lemma (Lemma 1), we have
Hence, using Lemma 2(b), we get
Now, assume that Y is an arbitrary separable Banach space. Take a nonzero continuous linear functional \(y^*\in Y^*\). Since the set-valued map \(x\mapsto \overline{y^*(G(x))}\), \(x \in I\), is convex and has closed values in \({\mathbb R}\), by the previous step,
Take arbitrary \(z\in \int _{\Omega } G\big (X_2(\omega )\big ) dP\). By the definition of the Aumann integral, there exists an integrable selection \(g\circ X_2\) of the set-valued map \(G\circ X_2\) such that \(z= \int _{\Omega } g\big (X_2(\omega )\big ) dP\). Using (4), we obtain
Since G is integrable bounded and the values \(y^*\Big (G\big (X_1(\omega )\big )\Big )\) are convex, by Lemma 2(c), we get
Since this condition holds for arbitrary \(y^*\in Y^*\) and, by Lemma 2(a) the set \(\overline{\int _{\Omega } G\big (X_1(\omega )\big ) dP}\) is convex and closed, by the separation theorem (see [16], Corollary 2.5.11), we obtain
and hence, using once more Lemma 2(c),
Consequently,
which finishes the proof. \(\square \)
Remark 5
In the above proof we use the Ohlin lemma (Lemma 1) to obtain the inequalities (3). Replacing in Theorem 4 the assumptions on \(X_1\) and \(X_2\) (the same as in Ohlin’s lemma) by any weaker conditions sufficient for (3) (for instance necessary and sufficient conditions such as in the Levin–Stečkin theorem [7]; cf. also [11]), we can obtain more general result. However, it should be emphasized that the assumptions in the Ohlin lemma are very convenient because they are simple and can be easy verified.
4 Applications
In this section, we present an application of the Ohlin-type lemma to obtain various inclusions related to convex set-valued maps in a simple and unified way. Some of these results (Corollaries 6–10) are known, but we present alternative proofs of them.
The first result is a counterpart of the classical integral Jensen inequality.
Corollary 6
(cf. [8]). Let \(G:I\rightarrow cl(Y)\) be integrable bounded set-valued map and \((\Omega ,\mathcal {A}, P)\) be a probability space with a non-atomic measure P. Then G is convex if and only if
for every integrable random variable \(X:\Omega \rightarrow I.\)
Proof
Assume first that \(G:I\rightarrow cl(Y)\) is a convex integrable bounded set-valued map and \(X:\Omega \rightarrow I\) is an integrable random variable. Take a random variable \(X_1:\Omega \rightarrow I\) with the distribution \(\mu _{X_1} = \delta _{\mathbb {E}[X]}.\) Then the distribution functions \(F_X, F_{X_1}\) satisfy condition (1) and \(\mathbb {E}[X]=\mathbb {E}[X_1]\). Therefore, by Theorem 4,
Now, assume that G satisfies condition (7) with every integrable random variable \(X:\Omega \rightarrow I.\) Fix \(x_1 , x_2 \in I\) and \(t\in (0,1)\), and take a random variable \(X:\Omega \rightarrow I\) with the distribution \(\mu _{X} = t \delta _{x_1} +(1-t)\delta {x_2}.\) Then \(\int _{\Omega } X(\omega )dP = tx_1 +(1-t)x_2\) and \(\int _{\Omega } G\big (X(\omega )\big ) dP = tG(x_1) +(1-t)G(x_2).\) Therefore by (7)
which proves that G is convex. \(\square \)
If in the above corollary we take a random variable X with the distribution \(\mu _{X} = t_1\delta _{x_1} + \cdots + t_n \delta _{x_n},\) where \(x_1,\ldots ,x_n \in I \) and \(t_1,\ldots ,t_n >0\) are such that \(t_1+\cdots +t_n =1\), then we obtain a counterpart of the discrete Jensen inequality.
Corollary 7
(cf. [10]). If a set-valued map \(G:I\rightarrow cl(Y)\) is convex and integrable bounded, then
for all \(n\in {\mathbb N}\), \(x_1,\ldots ,x_n \in I \) and \(t_1,\ldots ,t_n >0\) with \(t_1+\cdots +t_n =1.\)
We have also the following converse Jensen inclusion for convex set-valued maps.
Corollary 8
(cf. [6]). Let \(m, M \in I\), \(m<M\). If \(G:I\rightarrow cl(Y)\) is convex and integrable bounded, then
for all \(x_1,\ldots ,x_n \in [m, M] \), \(t_1,\ldots ,t_n >0\) with \(t_1+\cdots +t_n =1\) and \(\bar{x}= t_1x_1+\cdots +t_n x_n .\)
Proof
Take random variables \(X_1, X_2:\Omega \rightarrow I\) with the distributions
Then the distribution functions \(F_X, F_Y\) satisfy condition (1) and
Moreover
and
Therefore, by Theorem 4, we obtain (8). \(\square \)
The next two corollaries are versions of the Hermite–Hadamard inequalities for convex set-valued maps.
Corollary 9
(cf. [9, 17]). If \(G:I\rightarrow cl(Y)\) is convex and integrable bounded, then
for all \(a, b \in I,\ a < b.\)
Proof
Let \(X_1, X_2:\Omega \rightarrow I\) be random variables with the distributions \(\mu _{X_1} = \delta _{(a+b)/2}\), \(\mu _{X_2} = \frac{1}{2}(\delta _a + \delta _b)\) and let \(X_3:\Omega \rightarrow I\) has the uniform distribution on [a, b]. Then the pairs \(X_1, X_3\) and \(X_3, X_2\) satisfy the assumptions of Theorem 4. Moreover,
and
Therefore, by Theorem 4, we obtain (9). \(\square \)
Corollary 10
(cf. [9]) If \(G:I\rightarrow cl(Y)\) is convex and integrable bounded, \([a,b]\subset I\) and \(\mu \) is a Borel measure on [a, b] with \(\mu ([a,b])>0\), then
where \(x_{\mu }=\frac{1}{\mu ([a,b])} \int _a^b x \, d\mu (x)\) is the barycenter of \(\mu \) on [a, b].
Proof
By the mean value theorem \(x_{\mu } \in [a,b]\). Let \(X_1, X_2, X_3:\Omega \rightarrow [a,b]\) be random variables with the distributions
Then the pairs \(X_1, X_3\) and \(X_3, X_2\) satisfy the assumptions of Theorem 4. Moreover,
and
Therefore, by Theorem 4, we obtain (10). \(\square \)
The next two corollaries are counterparts for convex set-valued maps of the following inequalities concerning convex functions \(f: I \rightarrow \mathbb {R}\) (cf. [5, 15]):
for all \(a, b, c, d \in I \) such that \(a<c<d<b\),
and the Popoviciu inequality
for all \(x,y,z \in I\).
Corollary 11
If \(G:I\rightarrow cl(Y)\) is convex and integrable bounded, then
for all \(a, b, c, d \in I \) such that \(a<c<d<b\).
Proof
Let \(X_1, X_2:\Omega \rightarrow I\) be random variables with the distributions
Then the pair \(X_1, X_2\) satisfies the assumptions of Theorem 4. Moreover,
Therefore, by Theorem 4, we obtain (11). \(\square \)
Corollary 12
If \(G:I\rightarrow cl(Y)\) is convex and integrable bounded, then
for all \(x,y,z \in I\).
Proof
Let \(X_1, X_2:\Omega \rightarrow I\) be random variables with the distributions
Then the pair \(X_1, X_2\) satisfies the assumptions of Theorem 4. Moreover,
Therefore, by Theorem 4, the corollary is proved. \(\square \)
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Nikodem, K., Rajba, T. Ohlin-Type Theorem for Convex Set-Valued Maps. Results Math 75, 162 (2020). https://doi.org/10.1007/s00025-020-01292-3
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DOI: https://doi.org/10.1007/s00025-020-01292-3